Kurtosis of the order parameter in statistical physics
The Binder parameter or Binder cumulant[1][2] in statistical physics, also known as the fourth-order cumulant is defined as the kurtosis of the order parameter, s, introduced by Austrian theoretical physicist Kurt Binder. It is frequently used to determine accurately phase transition points in numerical simulations of various models. [3]
The phase transition point is usually identified comparing the
behavior of as a function of the temperature for different values of the system size . The transition temperature is the unique point where the different curves cross in the thermodynamic limit. This behavior is based on the fact that
in the critical region, , the Binder parameter behaves as , where .
Accordingly, the cumulant may also be used to identify the universality class of the transition by determining the value of the critical exponent of the correlation length. [1]
In the thermodynamic limit, at the critical point, the value of the Binder parameter depends on boundary conditions, the shape of the system, and anisotropy of correlations.
[1][4][5][6]
^ abcBinder, K. (1981). "Finite size scaling analysis of ising model block distribution functions". Zeitschrift für Physik B: Condensed Matter. 43 (2): 119–140. Bibcode:1981ZPhyB..43..119B. doi:10.1007/bf01293604. ISSN 0340-224X. S2CID 121873477.
^Binder, K. (1981-08-31). "Critical Properties from Monte Carlo Coarse Graining and Renormalization". Physical Review Letters. 47 (9): 693–696. Bibcode:1981PhRvL..47..693B. doi:10.1103/physrevlett.47.693. ISSN 0031-9007.
^K. Binder, D. W. Heermann, Monte Carlo Simulation in Statistical Physics: An Introduction (2010) Springer
^Kamieniarz, G; Blote, H W J (1993-01-21). "Universal ratio of magnetization moments in two-dimensional Ising models". Journal of Physics A: Mathematical and General. 26 (2): 201–212. Bibcode:1993JPhA...26..201K. doi:10.1088/0305-4470/26/2/009. ISSN 0305-4470.
^Chen, X. S.; Dohm, V. (2004-11-30). "Nonuniversal finite-size scaling in anisotropic systems". Physical Review E. 70 (5): 056136. arXiv:cond-mat/0408511. Bibcode:2004PhRvE..70e6136C. doi:10.1103/physreve.70.056136. ISSN 1539-3755. PMID 15600721. S2CID 44785145.
^Selke, W; Shchur, L N (2005-10-19). "Critical Binder cumulant in two-dimensional anisotropic Ising models". Journal of Physics A: Mathematical and General. 38 (44): L739–L744. arXiv:cond-mat/0509369. doi:10.1088/0305-4470/38/44/l03. ISSN 0305-4470. S2CID 14774533.
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